On completeness of the root functions of polynomial pencils of ordinary differential operators with constant coefficients

This article discusses the pencil of ordinary differential operators generated on $[0,1]$ by a linear differential expression of $n$-th order with constant coefficients, polynomially depending on spectral parameter $\lambda$ , and the two-point boundary conditions of a general form with coefficients which are polynomials of the spectral parameter $\lambda$.
It is assumed that the differential expression is homogeneous, the roots of its characteristic equation are distinct and different from zero, that is a fundamental system of solutions of the corresponding differential equation are pure exponent.
Provides definitions of $m$-fold completeness of the system of derived $m$-chains in the space of square integrable functions on the interval $[0,1]$.
The problem of finding sufficient conditions on the coefficients of the pencil, when there is an $m$-fold completeness of the system of the derived $m$-chains, is solved. A detailed history overview of the problem is given. It shows the urgency of solving this problem.
Then a characteristic polygon of the characteristic determinant of the pencil is introduce and on its basis the geometric classification of the pencils is given. Regular, almost regular, weakly and strongly irregular pencils are defined.
The method of obtaining sufficient conditions for multiple completeness of the root functions is to use a special solution of basic differential equation depending on an arbitrary parameter vector. We investigate important for the further features of this special solution, namely: Lemma of linear independence of a set of such solutions depends on linearly independent set of parameter vectors; Lemma of the characteristic polygons of special solutions when as vectors parameters are taken vectors constructed on the basis of the coefficients of the boundary conditions and the roots of the characteristic polynomial.
Sufficient conditions for multiple completeness formulated in terms of the existence of a sufficiently rich set of parameter vectors, which enables the scheme of the proof of multiple completeness, dating back to the famous work of Keldysh M.V. in 1951.
Key words: pencil of ordinary differential operators, root functions, eigen- and associated functions, multiple completeness, sufficient conditions of completeness, constant coefficients of differential expression, arbitrary location the roots of the characteristic polynomial, arbitrary two-point boundary conditions, nonregular pencil.